# Implementing a Fast Fourier Transform for Option Pricing

So, I'm in need of some tips regarding a small project I'm doing. My goal is an implementation of a Fast Fourier Transform algorithm (FFT) which can be applied to the pricing of options.

First concerns:

-which FFT, there are a lot of differents Algorithms, which can be called FFT, the most famous one being Cooley-Tukey I guess.

My thoughts on this: I prefer the most simple one, since this is no thesis or a big project, just a course on Algorithms. But it has to be compatible with option pricing (in contrast with the most -well in our general literature- referenced application of images/sound processing). So it depends on the form of input that is provided (on which I need some advice). I'm familiar with the several improvements, like a Fractional FFT, mixed radix FFT etc. But these seem pretty complex and optimization/performance driven, which is not relevant for my project.

-which Pricing model:

I Guess Black Scholes is a bit too 'flat' and I am aware of the several models that emerged after BS. So with the same objectives as stated above I'd initially prefer the Heston model.

There are a lot of considerations, and truth is I just can't see the wood for the trees.

Some background info:

My background is a B.Sc in Mathematics (Theoretical), so I have some understanding of fourier transforms. Goal is a working FFT implementation for caclulating option pricing.

(!)It does not have to be the fastest (no extreme optimization). Goals are trying to understand the chosen FFT and having a real-world working application.

So could you give some advice on the choices?

I've read a lot of papers on FFT + Option pricing, say all the decent hits on googles first few pages. But those studies were written with a much 'higher' cause.

-

I would say

• Start with Black Scholes to look at accuracy. In particular, you have a closed formula and you know what the characteristic function for lognormal is. Running FFT and comparing FFT pricing with the closed formula will give you an idea of what are the convergence issues, what is the behaviour at the boundaries (extreme strikes) etcetera.

• Then step forward to Heston with no correlation. In this case many people don't even bother with FFT but use Gauss-Legendre (or Gauss-Hermite or Gauss-Lobatto). Again, running FFT versus these alternative methods will teach you some lessons about convergence etcetera.

• Finally try Heston with correlation and some simple Levy model like Variance Gamma for example and compare with a PDE implementation.

In my opinion a pricing algorithm is worth talking about only if it is superior to all the others. That's why if I look at an algorithm the first think I want to know is how it compares with the others, what are its limits and its strenghts.

-

This note may be helpful you. Especially, chapter 4 in the note covered the FFT method for option pricing, the author also gave the algorithm and MATLAB code.

-

This is public knowledge what you need is a good book on how option strategies are built and used...

Heres some good starting points/books in which you can get a good framework to start building and applying your FFT option pricing method of choice

Options futures and other Derivatives

Options Volatility and Pricing

-